# Solve for: integral of (sqrt(x^3)-\sqrt[3]{x})/(6\sqrt[4]{x)} x

## Expression: $\int{ \frac{ \sqrt{ {x}^{3} }-\sqrt[3]{x} }{ 6\sqrt[4]{x} } } \mathrm{d} x$

Use the property of integral $\begin{array} { l }\int{ a \times f\left( x \right) } \mathrm{d} x=a \times \int{ f\left( x \right) } \mathrm{d} x,& a \in ℝ\end{array}$

$\frac{ 1 }{ 6 } \times \int{ \frac{ \sqrt{ {x}^{3} }-\sqrt[3]{x} }{ \sqrt[4]{x} } } \mathrm{d} x$

$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-\sqrt[3]{x} }{ \sqrt[4]{x} } } \mathrm{d} x$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-{x}^{\frac{ 1 }{ 3 }} }{ \sqrt[4]{x} } } \mathrm{d} x$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 6 } \times \int{ \frac{ x\sqrt{ x }-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$

Use $\sqrt[n]{{a}^{m}}={a}^{\frac{ m }{ n }}$ to transform the expression

$\frac{ 1 }{ 6 } \times \int{ \frac{ x \times {x}^{\frac{ 1 }{ 2 }}-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$

Calculate the product

$\frac{ 1 }{ 6 } \times \int{ \frac{ {x}^{\frac{ 3 }{ 2 }}-{x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$

Separate the fraction into $2$ fractions

$\frac{ 1 }{ 6 } \times \int{ \frac{ {x}^{\frac{ 3 }{ 2 }} }{ {x}^{\frac{ 1 }{ 4 }} }-\frac{ {x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$

Simplify the expression

$\frac{ 1 }{ 6 } \times \int{ {x}^{\frac{ 5 }{ 4 }}-\frac{ {x}^{\frac{ 1 }{ 3 }} }{ {x}^{\frac{ 1 }{ 4 }} } } \mathrm{d} x$

Simplify the expression

$\frac{ 1 }{ 6 } \times \int{ {x}^{\frac{ 5 }{ 4 }}-{x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\frac{ 1 }{ 6 } \times \left( \int{ {x}^{\frac{ 5 }{ 4 }} } \mathrm{d} x-\int{ {x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x \right)$

Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral

$\frac{ 1 }{ 6 } \times \left( \frac{ 4{x}^{2}\sqrt[4]{x} }{ 9 }-\int{ {x}^{\frac{ 1 }{ 12 }} } \mathrm{d} x \right)$

Use $\begin{array} { l }\int{ {x}^{n} } \mathrm{d} x=\frac{ {x}^{n+1} }{ n+1 },& n≠-1\end{array}$ to evaluate the integral

$\frac{ 1 }{ 6 } \times \left( \frac{ 4{x}^{2}\sqrt[4]{x} }{ 9 }-\frac{ 12x\sqrt[12]{x} }{ 13 } \right)$

Distribute $\frac{ 1 }{ 6 }$ through the parentheses

$\frac{ 2{x}^{2}\sqrt[4]{x} }{ 27 }-\frac{ 2x\sqrt[12]{x} }{ 13 }$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\frac{ 2{x}^{2}\sqrt[4]{x} }{ 27 }-\frac{ 2x\sqrt[12]{x} }{ 13 }+C,& C \in ℝ\end{array}$

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